Evaluation of selected value-at-risk approaches in normal

Evaluation of Selected Value-at-Risk

Approaches in Normal and Extreme

Market Conditions

Dissertation submitted in part fulfilment of the

requirements for the degree of Master of Science in

International Accounting and Finance

at Dublin Business School

August 2014

Submitted and written by

Felix Goldbrunner

1737701

Declaration:

I declare that all the work in this dissertation is entirely my own unless the words have been

placed in inverted commas and referenced with the original source. Furthermore, texts cited

are referenced as such, and placed in the reference section. A full reference section is included

within this thesis.

No part of this work has been previously submitted for assessment, in any form, either at Dublin

Business School or any other institution.

Signed:…………………………

Date:…………………………...

Table of Contents

Table of Contents....................................................................................................................... 4

List of Figures ............................................................................................................................ 6

List of Tables.............................................................................................................................. 7

Acknowledgements.................................................................................................................... 8

Abstract ...................................................................................................................................... 9

1. Introduction .......................................................................................................................... 10

1.1 Aims and Rationale for the Proposed Research ................................................................. 10

1.2 Recipients for Research...................................................................................................... 10

1.3 New and Relevant Research............................................................................................... 10

1.4 Suitability of Researcher for the Research......................................................................... 11

1.5 General Definition.............................................................................................................. 11

2. Literature Review................................................................................................................. 14

2.1 Theory ................................................................................................................................ 14

2.1.1 Non-Parametric Approaches ........................................................................................... 14

2.1.2 Parametric Approaches ................................................................................................... 15

2.1.3 Simulation – Approach.................................................................................................... 29

2.2 Empirical Studies ............................................................................................................... 31

2.2.1 Historical Simulation....................................................................................................... 31

2.2.2 GARCH........................................................................................................................... 32

2.2.3 RiskMetrics ..................................................................................................................... 32

2.2.4 IGARCH.......................................................................................................................... 33

2.2.5 FIGARCH ....................................................................................................................... 34

2.2.6 GJR-GARCH .................................................................................................................. 34

2.2.7 APARCH......................................................................................................................... 35

2.2.8 EGARCH ........................................................................................................................ 36

2.2.9 Monte Carlo Simulation .................................................................................................. 36

3. Research Methodology and Methods................................................................................... 37

3.1 Research Hypotheses.......................................................................................................... 37

3.2 Research Philosophy .......................................................................................................... 39

3.3 Research Strategy............................................................................................................... 39

3.4 Ethical Issues and Procedure.............................................................................................. 45

Research Ethics........................................................................................................................ 45

3.5 Population and Sample....................................................................................................... 45

3.6 Data Collection, Editing, Coding and Analysis ................................................................. 49

4. Data Analysis ....................................................................................................................... 50

4.1 Analysis of the period from 2003 to 2013.......................................................................... 51

4.2 Analysis of the period from 2003 to 2007.......................................................................... 54

4.3 Analysis of the period from 2008 to 2013.......................................................................... 58

5. Discussion ............................................................................................................................ 61

5.1 Discussion .......................................................................................................................... 61

5.2 Research Limitations and Constrains................................................................................. 63

6. Conclusion............................................................................................................................ 64

Publication bibliography .......................................................................................................... 66

Appendix A: Reflections on Learning ..................................................................................... 73

Appendix B.I Oxmetrics Output Crisis Sample ....................................................................... 75

Appendix B.II Oxmetrics Output Pre-Crisis Sample ............................................................... 94

Appendix B.III Oxmetrics Output Full Sample ..................................................................... 114

Appendix C Oxmetrics Screenshots....................................................................................... 133

List of Figures

Figure 1: Distribution and Quantile....................................................................................................... 12

Figure 2: Daily Returns (CDAX) .......................................................................................................... 13

Figure 3: Histogram of Daily Returns (CDAX).................................................................................... 13

Figure 4: Volatility Overview (CDAX) ................................................................................................ 16

Figure 5: Correlogram of Squared Returns (CDAX): 1 year ................................................................ 17

Figure 6: Absolute Returns (CDAX)..................................................................................................... 27

Figure 7: Correlogram of Absolute Returns (CDAX); 2003-2014........................................................ 27

Figure 8: Autocorrelation of Returns to the Power of d........................................................................ 28

Figure 9: LR(uc) and Violations............................................................................................................ 41

Figure 10: Violation Clustering............................................................................................................. 43

Figure 11: Price Chart ........................................................................................................................... 46

Figure 12: Return Series........................................................................................................................ 47

Figure 13: Histogram, Density Fit and QQ-Plot ................................................................................... 47

Figure 14: VaR Intersections................................................................................................................. 63

List of Tables

Table 1: Non-rejection Intervals for Number of Violations x ............................................................... 42

Table 2: Conditional Exceptions........................................................................................................... 43

Table 3: Descriptive Statistics............................................................................................................... 48

Table 4: Descriptive Statistics Sub-Samples......................................................................................... 49

Table 5: Ranking (2003-2013) .............................................................................................................. 51

Table 6: Test Statistics (2003-2013)...................................................................................................... 53

Table 7: Ranking (2003-2007) .............................................................................................................. 55

Table 8: Test Statistics (2003-2007)...................................................................................................... 57

Table 9: Ranking (2008-2013) .............................................................................................................. 58

Table 10: Test Statistics (2008-2013).................................................................................................... 60

Table 11: Ranking Overview................................................................................................................. 61

Acknowledgements

I would like to thank my family for their support during the last year, without their support the

completion of this program and thesis would not have been possible. Also I wish to express

my gratitude to my supervisor Mr. Andrew Quinn, without his support and creative impulses

this thesis would not be as it is today.

Abstract

This thesis aimed to identify the approaches with the most academic impact and to explain

them in greater detail. Hence, models of each category were chosen and compared. The non￾parametric models were represented by the historical simulation, the parametric models by

GARCH-type models (GARCH, RiskMetrics, IGARCH, FIGARCH, GJR, APARCH and

EGARCH) and the semi-parametric models by the Monte Carlo simulation. The functional

principle of each approach was explained, compared and contrasted.

Test for conditional and unconditional coverage were then applied to these models and

revealed that models accounting for asymmetry and long memory predicted value-at-risk with

sufficient accuracy. Basis for this were daily returns of the German CDAX from 2003 to

2013.

1. Introduction

1.1 Aims and Rationale for the Proposed Research

Recalling the disastrous consequences of the financial crisis, it becomes apparent that the

risks taken by financial institutions can have significant influences on the real economy. The

management of these risks is therefore essential for the functioning of financial markets and

consequently for the performance of the whole economy. Legislators and regulators have

therefore set their focus on various risk-management frameworks and even derived capital

requirements in accordance with certain risk measures. The most prominent of these is the so

called value at risk (VaR) measure, which was developed by J.P. Morgan at the end of the 80s

and tries to identify the worst loss over a target horizon such that there is a low, prespecified

probability that the actual loss will be larger (Jorion 2007b).

Value at risk plays an important role in the risk management of financial institutions. Its

accuracy and viability, both in normal and more extreme economic climates, is therefore

desirable. Since its introduction, academics and practitioners have developed a vast number of

methods to determine VaR, all of which are based on different assumptions and perspectives.

The question of finding an approach that delivers accurate results in normal and extreme

market conditions therefore poses a problem.

The aim of this thesis is to solve this problem and to answer the question concerning the most

accurate approach to determine value at risk in both normal and more extreme market

conditions.

1.2 Recipients for Research

The main recipients of this research will be managers responsible for risk management in

financial institutions such as banks and hedge funds as well as other financial-service

providers. Since this thesis aims also to explain the various value at risk approaches in a

generally intelligible way, independent and less-sophisticated investors can also be numbered

among the recipients. Additionally, researchers in the academic area of risk management, who

developed the models that will be tested, will also be beneficiaries of this research.

1.3 New and Relevant Research

To analyze the various approaches to value at risk, this thesis will identify the most accurate

approaches according to literature and then test them in terms of accuracy under both normal

market conditions and crisis conditions. In this way, a ranking will be proposed which will

show the most suitable methods for calculating value at risk. Most especially, the comparison

between normal function and function in a time of crisis is new and relevant research which

has not been thoroughly discussed in previous literature. As a result, practitioners as well as

academic researchers can benefit from this research.

1.4 Suitability of Researcher for the Research

To conduct this research, the researcher needs to be confident in approaching and utilizing

both fundamental and advanced statistics. Deeper knowledge about capital markets is required

as well as an understanding of widely used risk-management techniques. Moreover, the

researcher should be experienced in working with current spreadsheet applications such as

Microsoft Excel© or numerical computing suits such as OxMetrics©. The researcher has the

required experience in all of these areas, evidenced through his undergraduate degree (Upper

Second Class Honours in BSc in Business Administration) at the Catholic University of

Eichstätt-Ingolstadt, where he has already conducted research on the new liquidity

requirements proposed in the new Basel III regulation and on contingent capital with regard to

its contribution to the stability of financial markets.

Moreover, this researcher’s current Master’s-level course in international accounting and

finance enhances his knowledge of risk management and capital markets. My working

experience in form of a bank internship will also facilitate my perspective on the chosen topic.

1.5 General Definition

Value at risk is risk metric which measures the market risk in the future value of an asset or

portfolio. It is therefore a measure of uncertainty of a portfolio’s profit and loss (P&L), i.e.,

returns. To measure this risk, the portfolio’s profit and loss deviations from an expected value

are needed. This factor is called volatility and is the standard deviation σ from an expected

value μ. When considering a portfolio of assets, the correlation of the assets within the

portfolio is also a critical factor. To derive all these factors, assumptions have to be made

about the assets profit and/or loss distribution (Alexander 2009).

Combining all these risk factors, the value at risk can be defined as:

Definition 1:

The worst loss over a target horizon such that there is a low, prespecified probability

(confidence level) that the actual loss will be larger (Jorion 2007a).

It is therefore possible to come to statements of the following form:

“With a certainty of X percent, the portfolio will not lose more than V dollars over the time

T.”

Mathematically, this is the pre-specified upper bound of the loss distribution, the 1-α quantile

(Emmer et al. 2013) :

��������

(��) = ����

(��) = inf{ℓ|����(�� ≤ ℓ) ≥ ��} (1.1)

where :

�� = ��������

or when considering the whole P&L distribution the pre-specified lower bound, the α quantile

(Acerbi & Tasche 2002):

��������

(��) = ����

(��) = sup{��|����(�� ≤ ��) ≤ ��} (1.2)

where:

��

= ������������ ���������������� �������������������� ��ℎ�� ������������ ���������� ���� ������������ ���� ��������; ��������������

To illustrate this, Figure 1 depicts the distribution of asset returns and highlights the alpha

quantile.

Figure 1: Distribution and Quantile

To measure these returns, there are two possibilities: the arithmetic and the geometric rate of

return.

The arithmetic returns are compromised by the capital gain ���� − ����−1 plus interim payments

���� and can be defined as follows (Jorion 2007a):

���� =

���� + ���� − ����−1

����−1

(1.3)

Where:

���� = ���������� ���������� ���� �������� ��

����−1 = ���������������� ���������� ����������

���� = �������������� ����������������, ������ℎ ���� ������������������ ���� ��������������

Instead of this measurement, geometric returns also seem natural. These returns are expressed

in terms of the logarithmic price ratio, which has the advantage of not leading to negative

prices, as is mathematically possible with arithmetic returns:

���� = ln (

���� + ����

����−1

)

(1.4)

Figure 2 shows an example of logarithmic returns for the German Composite DAX (CDAX).

Figure 2: Daily Returns (CDAX)

When summing up these returns, another advantage is apparent. Other than the arithmetic

method, the geometric return of two periods is simply the sum of the individual returns. These

returns are sorted according to size and their frequency and result in the sampling distribution,

which is in practice often assumed to be a normal distribution.

Figure 3: Histogram of Daily Returns (CDAX)

To illustrate this historical distribution, a graphical analysis can be conducted in the form of a

histogram and the samples density function, as shown in Figure 3. It can be seen that the

sample distribution (red line) is somewhat similar to a normal distribution (green line). Most

especially, the tails (quantiles) differ significantly from the normal distribution in this

example.

For this reason, various approaches have been developed to estimate value at risk to cope with

these observations. These will be introduced in section 2.1 and the respective literature

reviewed in 2.2. Section three will then outline the research methodology and describe the

selected sample. Section four will present the findings of the research conducted and a

discussion will be provided in section five. Finally, section six will summarize the findings

and conclude.

2. Literature Review

The literature review will be organized as follows: First, the theory behind the selected

approaches will be explained. The second part will then consist of recent empirical studies

incorporating the explained approaches.

2.1 Theory

Manganelli & Engle (2001) classify existing value at risk approaches in three categories with

regard to their parameterization of stock price behavior: parametric, non-parametric, and

semi-parametric. This categories will be maintained for the analysis of approaches in the

thesis. At least one approach per category will be tested. Additionally, said categories will

also be maintained in this literature review.

2.1.1 Non-Parametric Approaches

According to Powell & Yun Hsing Cheung (2012), it is unlikely that stock returns follow a

parametric distribution, especially in times of a financial crisis. They therefore suggest the use

of non-parametric calculation methods, which will now be considered in more detail.

2.1.1.1 Historical Simulation

The most prominent and also the most straightforward method to calculate value at risk is

historical simulation.

Its simplicity lies in that no assumption about the population has to be made, since the actual

historical probability density function is used to derive VaR. The key assumption is therefore

that history repeats itself. A possible scenario in this framework is that an asset’s change in

market value from today to tomorrow will match an actual observed change that occurred

between a consecutive pair of dates in the past (Picoult 2002)

To calculate VaR, the nth

lowest observation has to be found, where n equals α of the

corresponding confidence level of the value at risk. In other words, the α-quantile has

to be determined as defined in (1.2).(Powell & Yun Hsing Cheung 2012).

Given a sample of 252 trading days, which equals about one year, the 95th percentile (α=0,05)

would be the 13th largest loss or lowest return. From this, it also follows that that the 99th

percentile will not be a constant multiple of the 95th percentile, and vice versa. Moreover, a

10-day VaR will not be a constant multiple of the one-day VaR. These limitations are a result

of not assuming independent and identically distributed (IID) random variables (Hendricks

1996).